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8   TOPO LOGICAL GROUPS       33

(If 0 is such that 0 < 0 < 1, then ||/(X) -/(x)||  \\f(6x') -f(8x)\\ - 2M(1 - 0),
and we can choose 0 so that d(6x, E  U) > i Vk2-l \\x'  x\\,

f(ex)\\^me\\X'-X\\   and    m6\\X'-x\\ -2M(1 -

8. TOPOLOGICAL GROUPS

If G is a group, in which the law of composition is written (for example)
multiplicatively, a topology on G is said to be compatible with the group
structure if the two mappings (x, y)t~* xy of G x G (endowed with the product
topology) into G, and x^x"1 of G into G, are continuous. A group endowed
with a topology compatible with its group structure is called a topological
group. We leave it to the reader to transcribe this definition (and all the
results which follow) into additive notation.

An isomorphism of a topological group G onto a topological group G' is by
definition an isomorphism of the group G onto the group G' which is bicon-
tinuous. If G' = G, we say automorphism instead of isomorphism.

If G is any group, the law of composition (x9y)t-+yx defines a group
structure on the set G (and this structure is different from the given one
unless G is commutative). The group so defined is denoted by G and is
called the opposite of G. Any topology which is compatible with the group
structure of G is also compatible with that of G, and hence makes G into
a topological group. The mapping xt-^-x'1 is then an isomorphism of G
onto G.

Examples

(12.8.1) The discrete topology and the chaotic topology (12.1.1) are com-
patible with the structure of any group. The topology of a normed space (in
particular R or C) is compatible with its additive group structure. On the
additive group Q of rational numbers, the topology defined by the /?-adic
distance d (3.2.6) is compatible with the group structure, because by virtue of
the definition of this distance and (3.2.6.4) we have

d(x0 +yQ,x + y) Max(rf(*0 , x), d(y0 , y))
and

If E is a (real or complex) Banach space and GL(E) the set of linear homeo-
morphisms of E onto E, then the topology induced on GL(E) by that of
; E) is compatible with the group structure ((5 J.5) and (8.3.2)).ed in U. If L is the closed